Table Of ContentCharge and momentum transfer in supercooled melts:
Why should their relaxation times differ?
Vassiliy Lubchenko
Department of Chemistry, University of Houston, Houston, TX 77204-5003
(Dated: February 6, 2008)
7 Thesteadystatevaluesoftheviscosityandtheintrinsicionic-conductivityofquenchedmeltsare
0 computed,intermsofindependentlymeasurablequantities. Thefrequencydependenceoftheacdi-
0 electricresponseisestimated. Thediscrepancybetweenthecorrespondingcharacteristicrelaxation
2
times is only apparent; it does not imply distinct mechanisms, but stems from theintrinsic barrier
n distribution for α-relaxation in supercooled fluids and glasses. This type of intrinsic “decoupling”
a is argued not to exceed four orders in magnitude, for known glassformers. We explain the origin
J of the discrepancy between the stretching exponent β, as extracted from ǫ(ω) and the dielectric
2 modulusdata. Theactualwidthofthebarrierdistributionalwaysgrowswithloweringthetemper-
1 ature. Thecontraryisanartifact ofthelargecontributionofthedc-conductivitycomponenttothe
modulus data. The methodology allows one to single out other contributions to the conductivity,
] as in “superionic” liquids or when charge carriers are delocalized, implying that in those systems,
n
charge transfer does not require structural reconfiguration.
n
-
s
i I. INTRODUCTION fromsystemtosystem: Forinstanceinmoltennitrates,it
d
is aboutfour ordersof magnitude atT [11], while in sil-
. g
t ver containing superionic melts, the τ /τ ratio becomes
a Molecular motions in deeply supercooled melts and s σ
m glasses are cooperative so that transporting a single as large as 1011 [12], i.e. almost as much as the whole
dynamical range accessible to the melt! This disparity
- moleculerequiresconcurrentrearrangementofuptosev-
d eral hundreds of surrounding molecules. Such high de- suggestedthatthemechanicalrelaxationandthe electri-
n calconductivity in these systems werein fact due to dis-
gree of cooperativityresults in highbarrierseven for the
o tinctmechanisms: Athighertemperatures,thetimescale
smallest scale molecular translations. These high barri-
c
separationis small so that the two processesstrongly af-
[ ersunderlytheslow,activateddynamicsindeeplysuper-
cooled melts and the emergence of a mechanically stable fect each other, or “mix”, while at lower temperatures,
1 the processes become increasingly “decoupled” [13]. At
aperiodiclattice,ifameltisquenchedsufficientlyrapidly.
v suchlow temperatures,the mechanicalrelaxationoccurs
The Random First Order Transition (RFOT) method-
5
via the aforementioned, activated concerted events, also
6 ology, developed by Wolynes and coworkers, provides a
calledtheprimary,orα-relaxation. Otherprocessesthat
2 constructive microscopic picture of the structural rear-
1 rangements in supercooled melts and quenched glasses. seemto decouplefromthe mechanicalrelaxationinclude
0 TheRFOThasquantitativelyexplainedorpredictedthe nuclear spin relaxation, rotational diffusion, and the dif-
7 fusion of small probes. (For reviews, see [13, 14, 15])
signaturephenomena accompanyingthe glasstransition,
0
including the connection between the thermodynamic Here we focus on two specific transport phenomena:
/
t and kinetic anomalies [1, 2, 3], the length scale of the low-frequency momentum transfer, i.e. the viscous re-
a
m cooperative rearrangements [2], deviations from Stokes- sponse, and the ionic conduction in supercooled melts.
Einstein hydrodynamics [4], aging [5], the low tempera- Notwithstandingthecomplicationsneededtoanalyzethe
-
d ture anomalies [6, 7, 8], and more. (See [9] for a recent electrical modulus data [16, 17, 18, 19, 20, 21, 22], the
n review.) mismatch between the typical relaxation times, corre-
o Perhaps the most dramatic experimental signature of spondingtothetwotypesoftransport,isclearlypresent.
c
the glass transition is the rapid super-Arrhenius growth Furthermore, in the case of superionic compounds, one
:
v of the relaxation times with lowering the temperature, may show (see below) that conduction occurs without
i from about a picosecond, near the melting point T , to distortingtheliquid’sstructurebeyondthetypicalvibra-
X m
as long as hours, at the glass transition temperature T . tional displacements. This is much less obvious for com-
g
r
a These relaxation times are deduced via several distinct pounds where the ionic motions are “decoupled” from
experimentalmethodologiesandalldisplayanextraordi- the bulk structural relaxation by four orders of magni-
narily broad dynamical range. Nevertheless, making de- tude or less, the latter dynamic range comparableto the
tailedcomparisonsbetweenthose distinct methodologies breadth of the α-peaks in dielectric dispersion in insu-
has required additional phenomenological assumptions. lating melts near T (see e.g. [23]). Accounting for the
g
Mysteriously, these comparisons show a significant de- distribution width is essential here because the viscosity
gree of mismatch, sometimes by severalordersof magni- and conductivity are distinct, in fact exactly reciprocal
tude. For example, the phenomenological “conductivity types of response: In momentum transfer, the velocity
relaxation time” τ [10], is consistently shorter than the gradient is the source, and the passed-on rate-of-force is
σ
mechanicalrelaxationtimeτ ,especiallyatlowertemper- the response. In charge transfer, the force on the ion is
s
atures. The apparent time scale separation varies wildly thesource,whilethearisingvelocityfieldistheresponse.
2
Consistent with this general notion, in computing the ξ
F(N)
viscosity, we will average the relaxation time, with re-
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The microscopic calculation and comparison, of the
viscosity and the ionic conductivity, are thus the main FIG. 1: (a): Typical nucleation profile of one aperi-
focus of this article. The two quantities are computed, odic lattice, within another, in deeply supercooled liquids.
intermsofthebarrierdistributionandothermeasurable N ≡ (4π/3)(r/a)3, N‡ is the typical transition state size:
materialproperties,inSectionsIIandIIIrespectively. To (dF/dN)N‡ = 0. ξ is the volumetrically defined cooperative
length: N∗ ≡ (ξ/a)3, where F(N∗) = 0. (b): Cartoon of a
performcomparisonswithexperimentandassesstheup-
structural rearrangement. The shown magnitude of ξ corre-
per limit on the “inherent decoupling” between the two
phenomena, we will discuss the barrier distribution in sponds to a temperature near Tg on 1 hour scale. The two
setsofcircles-solidanddashedones-denotetwoalternative
some detail, in Section IV. We will find that indeed, the structural states. dL ≃ a/10, or the “Lindemann length”, is
degreeofdecouplingshouldincreasewiththewidthofthe thetypical bead displacement duringa transition.
barrierdistribution,andhenceatlowertemperatures,as
demonstratedbytheRFOTmethodology[3]. Wewillas-
sessthedeviationbetweenα-relaxationtimes,asdeduced T is related to the meanfield temperature T but is al-
cr A
from viscosity, ionic conductance, and the maximum in ways smaller. The bead size a may be unambiguously
ǫ′′(ω). Further,wewillexemplifypotentialambiguitiesin determinedfromthe fusionentropyofthe corresponding
using the dielectric modulus formalismin estimating the crystal, when the latter entropy is known [27], or else
relaxationtimedistribution. Thelattertechiquehassug- can be computed from the fragility D using the univer-
gestedthatforsomesubstances,thedistributionwidthin salrelationshipbetween the latter andthe heatcapacity
fact decreases with lowering the temperature, in conflict jump at T : D = 32/∆c , as derived in RFOT [2, 28].
g p
with the present results and the correlationbetween the Alternatively, if the configuration entropy can be reli-
stretchingexponentβ andtemperature,predictedearlier ably estimated, one may use the RFOT-derived relation
by the RFOT theory [3]. We will see that the conflict is for the configurational entropy per bead s .8k [2],
c B
artificialandresultsfromthelargecontributionoftheac- which is somewhat sensitive to the barrier-so∼ftening ef-
component to the modulus data, consistent with earlier, fects though [27].
phenomenological arguments [17, 24].
Once locally metastable, the liquid may reconfigure
butinanactivatedfashion,i.e. bynucleatinganewape-
riodic structure within the present one. Such activated
II. VISCOSITY events occur, on average, once per typical α-relaxation
time τ, per region of size ξ. The nucleus grows in a
ThekeymicroscopicnotionbehindtheRFOTmethod- sequence of individual, nearly-barrierless bead moves of
ology is that, regardless of the detailed interparticle po- length dL a/10 [25, 29] and time τmicro 1 ps [29].
≃ ≃
tentials,localaperiodicarrangementsinclassicalconden- The overallsequence of elemental moves typically corre-
satesbecomemetastablebelowacertaintemperatureT sponds to the following activation profile, see Fig.1(a):
A
(oraboveacertaindensity)[25,26]. Chemicaldetailand
molecular structure affect the value of TA, and the vis- F(N)=γ√N −TscN, (1)
cosity of the fluid. If the viscosity is high enough, one
may cool the liquid so that it becomes locally trapped where N is the size of a reconfigured region. The “sur-
inmetastableminima, while avoidingthe nucleationofa face term” γ√N is the mismatch penalty for creating
periodic crystal, which would have been the lowest free one aperiodic structure within another. sc is the excess,
energy state. Another system-dependent quantity is the “configurational” entropy of the liquid per bead, hence
size a of the elemental structural unit in the metastable the entropic,bulk term( TscN), whichdrivesthe tran-
−
liquid, or “bead”: the length a plays the role of the lat- sition and reflects the multiplicity of possible aperiodic
ticespacingintheaperiodicstructure,andisindeedquite arrangementsinaregionofsizeN. Themaximumofthe
analogous to the size of the unit cell in an oxide crystal, profile:
or it may correspond to the size of a rigid monomer or
side chain in a polymer. The size a characterizes the γ2
F‡ =max F(N) = (2)
range of the local chemical order that sets in during a { } 4s T
c
crossover,atatemperatureT ,fromcollisiondominated
cr
transport to activated dynamics [27]. The temperature isachievedatN‡ =N∗/4,whereF(N∗)=0,sothatthe
3
typical relaxation time is wherewe haveremovedthe prime atτ, implying averag-
ing with respect to the actual barrier distribution. The
τ =τ eF‡/kBT τ eDTK/(T−TK). (3)
micro micro equation above can be rewritten as
≡
Thisformulaworkswellatτ >1nsecorso. Theformon
2k T k T
ther.h.s. istheVogel-Fulcherlaw,derivedinthe RFOT. η =(a/d )2 B τ 60 B g τ , (5)
The end result of a cooperative, activated event is a L πa3 h i≃ a3 h i
reconfigured region of size ξ, where each of the N∗
(ξ/a)3 beads has moved the Lindemann length d , o≡r since the Lindemann ratio, dL/a ≃ 0.1 has been argued
L to change at most by 10% between T and T [29].
so, see Fig.1(b). Both ξ and the nucleation critical size, m g
Another instructive way to present Eq.(4) is to note
r‡ = ξ/41/3, increase with lowering the temperature,
that the Lindemann length is nearly equal to the typi-
roughly as r‡ ξ 1/(T T )2/3 [1, 2, 27]. Here,
TK is the so c∝alled∝ideal gla−ss Ktransition temperature, cwailthaimnp5l%ituodresoof,hsiegeh-Ffrige.q3ueonfcRyevf.i[b2r9a]t.ioTnhs:edvLib≃ratdiovinbar,l
where the excess liquid entropy s , extrapolated below
c amplitude is fixed by the equipartition theorem, since
T , would vanish [30]. At T , ξ is still quite modest,
g g perbead: K a3(d /a)2 =k T,whereK isthehigh-
only about six beads across [2, 6]. Activated transport ∞ vibr B ∞
frequency elastic constant of the aperiodic lattice. One
becomes dominant below the temperature T , such that
cr gets, as a result, a Maxwell-type expression:
r‡(T ) = a, which corresponds, apparently universally,
cr
toτ/τ 103,orviscositiesontheorderof10Ps[27].
micro η K τ . (6)
At times sh≃orter than 103τ , one may then speak of ∼ ∞h i
micro
a local aperiodic lattice on length scales of the cooper- The last equation provides an easy way to see that the
ativity size ξ, since the slow structural reconfigurations estimates in Eqs.(4)-(6) agree well with the experiment:
have now time-scale separated from the vibrations [27]. Judging from the sound speed in glasses [33, 34], the
Becauseofthe localnatureofstructuralrelaxations,one typicalhighfrequencyelasticmodulusisabout109 1010
speaks of dynamic heterogeneity, or a “mosaic” of coop- Pa, i.e. comparable but somewhat less than th−ose of
erative rearrangments[2]. The heterogeneityis two-fold: crystals. The range of relaxation times 10−12 104 sec,
On the one hand, a local rearrangment implies that the implies 10−3 1013 Pasec for the viscosity, as−is indeed
surrounding structure is static during the transition, up observed. Alt−ernatively·, one may obtain these figures by
to vibrations. On the other hand, because of the spatial substituting a typical a 3˚A (see [27, 28] for specific
andtemporalvariationinthelocaldensityofstates(and estimates of bead sizes/de∼nsities).
hence variations in sc), local reconfigurations are gener- Finally, the exploited equivalence between the time
ally subject to somewhat different barriers in different and ensemble averages implies that crystallization has
regions [2] (see Eq.(2) and also Section IV). not begun during the experiment, of course. The latter
Computation of the viscosity in such a dynamically possibilityaddsuncertaintyintoviscositymeasurements,
heterogeneous environment may be done in two steps: as the presence of crystallites would greatly broaden the
First compute the viscosity in a medium with a homo- dynamic range of local relaxations owing to relatively
geneous relaxation time, call it τ′, and then average out slow crystal nucleation events and the slow hydrody-
with respect to the true distribution of the relaxation namics near the crystallites. Similarly, long chain mo-
times. This procedure is valid in view of the equiva- tions in polymeric melts would also introduce additional
lence of time and ensemble average. When the relax- long time scales into the problem. Our derivation does
ationrateisstrictlyspatiallyhomogeneous,onemayfor- not apply to those situations. We note that optical
mally define a diffusion constant for an individual bead: transparency, which is often used as an indicator of no-
D′ =d2L/6τ′, since a bead moves the Lindemann length, crystallinity,does not ensure that crystallites - hundreds
once per time τ′, on average. Note that since a bead’s of nanometers across or smaller - are absent. Therefore
movements, as embodied in dL and τ′, are dictated by a rigorousexperimental study should, in the least, check
its cage, this is an example of “slaved” motion, to bor- whether performing viscoelastic measurements has en-
row Frauenfelder’s adjective for conformational changes hanced the crystallization of the sample. Ideally, X-ray
of a protein encased in a stiff solvent [31, 32]. One may diffractionshouldbemonitoredinthecourseofviscosity
associate, by detailed balance, a low-frequency drag co- measurements.
efficient to that diffusion constant: ζ′ = k T/D′. Such
B
dissipative response implies irreversible momentum ex-
changebetweenachosenparticleanditshomogeneous(!) III. IONIC CONDUCTIVITY
surrounding, hence a Stokes’ viscosity η′ = ζ′/(6πa/2),
where a/2 is used for the radius of the regioncarvedout
In any supercooled melt, whether regarded ionic or
in the liquid by a single bead. Averaging with respect
not,the beadscarryanadditionalcharge,relativeto the
to τ′ yields for the steady state viscosity of the actual
corresponding crystal, because of the lack of crystalline
heterogeneous liquid:
symmetry. As a result,eachstructuralreconfigurationis
2kBT characterized by a transition-induced electric dipole, see
η = τ , (4)
πad2 h i Fig.2.
L
4
construction of liquid states [5]. Repeating that argu-
ment, but in the presence of electric field E, yields for
+ ζ q
dL the typical free energy profile for structural reconfigura-
tion in steady state:
− ζ q
F(N)=γ√N µ (N)E Ts N, (10)
T c c
− −
where N is the size of the rearrangedregion and
µ(in)
∆µ
mol
mol
N/2
ξ a µ(f) µ (N)= µ(f) µ(in) (11)
mol T j − j
Xj h i
FIG. 2: Shown on the left is a fragment of the mosaic of
istheoveralldipole changeinthatregion. Thesubscript
cooperatively reconfiguring regions in thesupercooled liquid.
“c” in E signifies that the latter is a cavity field. The
Expandedportionshowshowrotationofabondleadstogen- c
field dependent term µ (N)E k T is overwhelm-
eratinganelementaldipolechangeduringatransition,where | T c| ≪ B
thepartial charges on thetwo beads are ±ζq. inglysmallerthantheothertermsinEq.(10). Asaresult,
the transition state dipole moment is not field-induced,
but, again, is “slaved” to the lattice. More formally, one
ThelatterwasestimatedtobeaboutaDebyeorso,for mayusetheargumentfromRef.[6]showingthattheden-
most molecular substances [35]. This value comes about sityofstructuralstatesatthereconfigurationbottle-neck
as we may break up the whole domain into N∗/2 pairs, is of the order 1/T , implying the field will not affect
g
where is pair has an elemental dipole µ , i=1...N∗/2: the specific sequence of elemental moves, but will affect
i
µ =ζeq. Here q is the elementary charge: q =100e, thedynamicsmerelybyshiftingtheenergiesalongstruc-
i
h| |i
and ζ <1 characterizesthe excess charge. This quantity turally dictated sequences of moves. Thus in the lowest
ζ is usually small reflecting small deviations from the order in E , τ−1(E ) = τ−1(E = 0)(1+µ‡E /k T),
c c c T c B
crystallinesymmetry,inthecaseofmolecularcrystals,or yielding
reflectingtheweakinteractioninVanderWaalssystems.
Alternatively, the overalldensity of charged/polarbeads µ (µ‡E ) 1
may be low. Ionic melts, by the very meaning of the j = T T c , (12)
τ k T
* + B
term,aredistinctfrommolecular/VanderWaalssystems
in that nearly all beads are strongly charged, implying
where µ‡ µ (N‡) is the transition dipole moment at
ζ 1. Tobe morespecific, theconclusionsofthisarticle T ≡ T
∼ the zero-field transition state. The cavity field, see e.g.
willbe exemplifiedwithanoftenstudiedmixture of40%
[36], is related to the external field E(∞) by
Ca(NO ) -60%KNO (“CKN”), T 330K.
3 2 3 g
≃
During a transition, each dipole turns by an angle
3ǫ (ω)
(dL/a). The total transition dipole: Ec(ω)=E(∞)(ω) b , (13)
2ǫ (ω)+1
b
N∗/2
µ = µ(f) µ(in) (7) where ǫ (ω) is the dielectric constant of the surrounding
T j − j b
Xj h i bulk. Since a steady currentis implied in the derivation,
(the imaginarypart of) ǫ (ω) divergesat zero frequency,
b
scales as √N∗ because of the random orientation of the implyingE =E(∞)(3/2). Onethusobtainsfortheionic
c
elemental dipoles [35]: conductivity tensor:
µ ζ(qa)[(ξ/a)3/2]1/2(d /a). (8)
T ≃ L (µ ) (µ‡) 3
σ = T i T j . (14)
When the dipole density is uniform, every transition re- ij τ 2k Tξ3
* + B
sultsinalocalarrangementequallyrepresentativeofthe
liquid structure. In other words, structural transitions Bearing in mind that N‡ = N∗/4 = (ξ/a)3/4, and
donotmodify the overallpatternoftheimmediate coor-
that the liquid is isotropic, on average: (∆µ) (∆µ) =
i j
dination shell. Transitions lead to a local ionic currents: δ (∆µ)2/3, one finally has: h i
i′ = µ′ /τ′, per region of volume ξ3. In the presence of ij
T
an electric field, the net current density: ∆µ2 1
σ =δ mol , (15)
j = j′ = 1 µT , (9) ij ij 8a3kBT (cid:28)τ(cid:29)
h i ξ3 τ
D E where
is non-zero because the dipole moment at the transition
state is correlated with the overall transition dipole mo- ∆µ2 µ(f) µ(in) 2 (16)
ment. The latter can be shown using Wolynes’ library mol ≡ j − j
(cid:28)h i (cid:29)
5
is the average elemental dipole change squared. Finally stances, the dielectric constant even at very high fre-
notethatincovalentlynetworkedmaterials,wheredipole quences is significantly larger than unity. In CKN, for
assignmentmaybeambiguous,onemaystillestimatelo- instance, ǫ′( ) 7 [11], allowing us to write as before:
cal dipole changes using the known piezoelectric proper- E E(∞)(3∞/2)≃. One thus obtains, in a standard fash-
c
≃
ties of the corresponding crystal, see [35] for details. ion, for the frequency dependent dielectric constant in
the absence of macroscopic current:
The derivationabovedoesnotapply to systemswhere
the dipole density is significatly non-uniform. For in- ∆µ2 1
stance, glycerol has one polar, OH group per non-polar, ǫins(ω)−ǫ∞ =4πhp1p2i4a3kmoTl 1 iωτ , (17)
aliphaticgroup,implyingtheliquidisnon-homogeneous, B (cid:28) − (cid:29)
dipole moment wise, on the α-relaxation time scale. In where the label “ins” signifies the absence of dc conduc-
suchsystems,thepremisethatstructuralrearrangements tivity.
result in equally representative configurations of the liq- Inthe presenceofsteadycurrent,the fullresponseper
uiddoesnothold. Intheglycerolexample,ionicconduc- domain is the sum of the steady current from Eq.(15)
tion would imply breaking OH or CH bonds. In CKN, and the ac current from Eq.(17). The addition of the
on the other hand, the overallbond pattern, aroundany dc contribution, iσ/ω, to the full dielectric response will
atom, does not change significantly during a transition, increase the absolute value of ǫ (ω). This means that
b
even though individual bonds distort by the Lindemann Eq.(17) should work even better. One thus gets for the
length, as mentioned. Eq.(15) thus places the absolute full dielectric response of a conducting substance:
upper limit on the intrinsic ionic conductivity of a melt.
∆µ2 1 σ
By “intrinsic” we mean that the computed currents are ǫ(ω) ǫ =4π p p mol +i , (18)
− ∞ h 1 2i4a3k T 1 iωτ ω
always present in the fluid and result from the intrin- B (cid:28) − (cid:29)
sic activated transport: Local bond pattern does not where σ is the dc conductivity from Eq.(15).
change significantly in the course of an individual ac- One needs to know the distribution of the transi-
tivatedevent,butonlyinthecourseofmanyconsecutive tion energies E to estimate the quantity p p
1 2
events, since during an individual event, the molecular 1/4cosh2(E/2k T) . Since ξ is the smallesht posisib≡le
B
displacements barelyexceedtypicalvibrationaldisplace- size of a rearranging unit, these rearrangments corre-
ments. Conversely, if a system displays a higher con- s(cid:10)pond to the elemen(cid:11)tary excitations in the system. We
ductivity than prescribed by Eq.(15), one may conclude thus conclude, based on equipartition, that the typi-
thattheionmotiondoesnotrequirestructuralreconfigu- cal value of E is roughly k T, implying that p p is
B 1 2
ration. Here,thebondpatternactuallychanges,however close to its maximum value of one quarter buthis likiely
these arenotscaffoldbonds ofthe aperiodiclattice com- smaller by another factor of two or so. Assuming then,
prising the fluid (or glass). (More on this below.) for the sake of argument that p p = 1/8, one gets
1 2
To simplify comparison of Eq.(15) with experiment, ∆µ2mol 2(ǫ′ ǫ ), within an ohrderiof magnitude. By
let us express the combination of the bead charge ζq a3kBT ≃ 0− ∞
Eq.(15), this implies a Maxwell-likerelationbetween the
and size a, in Eq.(15), through the finite frequency di-
dc conductivity and the real part of the dielectric re-
electric response,a measurablequantity inprinciple (see
sponse:
below). Thelatteristheresponseofarearrangingregion
in the absence of bulk current, i.e. with a fixed environ- 1
σ (ǫ′ ǫ ) , (19)
ment, up to vibrations. It is convenient to choose such ∼ 0− ∞ τ
regions at volume ξ3, so that each region has two struc- (cid:28) (cid:29)
with animportantdistinction, though,that here one av-
tural states available, within thermal reach from each
erages the inverse relaxation time. The (ǫ′ ǫ ) dif-
other, separated by a barrier sampled from the actual 0 − ∞
ference in CKN, to be concrete, is about (20 7) = 13
barrier distribution in the liquid. If the two states, “1”
−
[11, 37]. This implies, by Eq.(17) and CKN’s T 330K
and “2” are characterized by dipole moments µ and g
1 ≃
µ respectively, the expectation value of the dipole mo- [37], that ζq 3e, at a=3˚A, a reasonable value for the
m2ent of the region is µ=(µ +µ )/2+∆µ(p p )/2, beadcharge.≃Note thata=3˚Ais consistentwith CKN’s
1 2 2 1
pwrhoebraeb∆itiµes≡to(µoc2c−upµy1s)t;apte1 1anadndp22=re(s1pe−ctpiv1e)−lya.reTthhee ∆[2]c.p ≃ .12 cal/g K [38] and the mentioned D = 32/∆cp
rδellna(tpive/pp)op=ula∆tiµonE(/pk2 −T.p1A) tderpeaenlidstsicofineltdhestfireenldgtvhisa, forNtohwe,csounbdsuticttuivtiintygσCKN3’s 1(ǫ0′0−−10ǫ∞τ)−1into(OEhqm.(1m5))−y1iesledcs.
2 1 c B ≃ ·
i.e. ∆µE /k T 1,onehas forthe field-inducedshift Naively replacing τ−1 with 1/ τ would imply, at the
of th|e relactivBe p|o≪pulation: δ(p p ) 2p p (∆µE ). glasstransition,where τ 102 h1(cid:10)i03se(cid:11)c,aconductivity
Similarly to the preceding arg2um−en1t, ≃(∆µ1) (2∆µ) c= of the order 10−13(cid:10) 10h(cid:11)−i14∼(Ohm−cm)−1, which is three
i j
(N∗/2)δ (∆µ )2/3. Further, since weh have frozenithe to four orders of m−agnitude below the observed value
ij mol
structural transitions in the surrounding region, in es- [11, 37]. Note that the value τ−1 τ = 103 104 is
h i −
timating the cavity field, one must use ǫ(ω) with the just the magnitude of decoupling observed in CKN near
(cid:10) (cid:11)
α-relaxation contribution subtracted. This does not T [11,13],andisinfactexpectedforafragilesubstance
g
introduce much ambiguity because in most ionic sub- such as CKN is, as we will argue in the following.
6
IV. BARRIER DISTRIBUTION AND THE
DECOUPLING 101
100
Relaxation barriers in supercooled liquids are dis-
F)10-1
tributed because the local density of states is non- p(
uniform, leading to variations in the local value of the 10-2
configurational entropy and hence the RFOT-derived 10-3 Eq.(21)
barrierfromEq.(2). In the simplest argument,the gaus- Eq.(23)
sian fluctuations of the entropy translate into gaussian 10-4 0.5 0.6 0.7 0.8 0.9 1
F
fluctuations in the barrier, where the relative deviations
of the two quantities, from the most probable value, are
FIG. 3: The barrier distributions from Eqs.(21) and (23).
given by [3]:
Theδ-functionportionofthedistributionfromEq.(21)isnot
e
δF δs 1 shown. δF =0.25.
c
δF = = , (20)
≡ Fmp sc 2√D
where D is the liequid’s fragility from Eq.(3). The quan-
for the lowest order, quadratic fluctuations of entropy
tity1/2√Dvariesbetween0.05and0.25orso,forknown suffices. The quantity τ τ−1 , that characterizes the
glassformers, the low and high limits corresponding to h i
apparent decoupling, computed with the XW’s distribu-
strong and fragile substances respectively. (cid:10) (cid:11)
tion, is shown with the dashed line in Fig.4, at T , as a
g
Xia and Wolynes (XW) further argued that the real
function of the relative distribution width δF.
barrier distribution should be cut-off at the most proba-
ble value because a liquidregionwithrelativelylow den- Howrobustisthepredictionbasedonthesimplefunc-
sity of states is likely neighbors with a relatively fast tional form for the barrier distribution fromeEq.(21)? In
region [3]. In addition we may recall that in the li- spite of its quantitative successes, one may argue that
braryconstruction,themostlikelyliquidstateistheone the true barrier distribution should be a smoother func-
where the liquid is guaranteed to have an escape trajec- tion, near F = 1. One way to see this is to computing
tory [5]. This means that the most probable barrier is ǫ′′(ω) from Eq.(17) via the distribution in Eq.(21): The
also the maximum barrier. One may conclude then that obtained cuerves are a sum of two peaks, one of which is
thenaiveGaussiandistributionisadequateatsmallbar- broader,onetheotherisnarrowerthantheexperimental
riers, but significantly overestimates the probability of ǫ′′(ω). The two peaks correspond to the half-Gaussian
barrierslargerthanthetypicalbarrier. Putanotherway, andthedelta-functioninEq.(21)respectively. Letussee
the trajectoriescorrespondingto higherthanmostprob- that knowing the precise formof the barrierdistribution
able barrier in the naive Gaussian, all contribute to the however is not essential in quantitative estimates of the
F F range. XW have implemented this notion by decouplingsolongasweaccountcorrectlyfortheoverall
mp
rep≤lacingthe r.h.s. ofthe simplestGaussiandistribution widthofthe distributionandits decayatthe lowbarrier
by a delta-function centered at F [3]: side.
mp
It is straightforwardto show that there exists a distri-
e−(1/Fe−1)2/2δFe2 1 butionthat(a)satisfiestheserequirementswithoutintro-
p (F)= + δ(F 1), (21)
1
2 − ducing adjustable constants, (b) reproduces the experi-
2π(δF)2F2
mental ǫ(ω) and does as well as the XW form for the β
e q e
vs. D correlation. As wehavealreadydiscussed, the low
where F ≡ F/Fmp < 1+e, aend we took advantage of barrier wing of the distribution in Eq.(21) is adequate.
the temperature-independence of the relative width in
On the other hand, the high barrer wing should include
Eq.(20)e. This approximate form does not use adjustable
thecontributionsfrombothsidesoftheoriginalGaussian
parameters and quantitatively accounts for the correla-
peak, which are both of width δF/2. “Stacking” these
tion between the fragility and the stretching exponent
two on top of each other, to the left of F , results in a
β [3], and the deviations from the Stokes-Einstein rela- mp
distributionofwidthδF/4(seeasloAppendix). Further,
tion. The distribution in Eq.(21) is shown in Fig.3. The
based on the known ǫ(ω) data, the barrier distribution
only difference of Eq.(21) with the XW’s form is that
shouldbe wellapproximatedbyanexponential,suggest-
they used a purely gaussian form for F < 1, whereas ing we use p(F) eF/(δF/4) near F . Indeed, this im-
mp
we follow their own suggestion and employ the more ac- ∝
plies p(τ) τ(4kBT/δF)−1. At frequencies not too close
curate F ∝ 1/sc (where sc is gaussiaenly distsributed tothe max∝imumofǫ′′(ω)andthe rapiddrop-offatsmall
of course). The accurate evaluation of the left wing of
F, one has an approximate power law:
the distribution is imperative in estimating the average
rate τ−1 e−F/kBT, because the latter is a rapidly vary-
micro
ing function of F. (k T is significantly less that δF at e ∞ ωτ
B ǫ′′(ω) dττ(4kBT/δF)−1 ω−4kBT/δF.
low temperatures.) Note that because of the rapid de- ≃ 1+(ωτ)2 ∝
Z0
cay of the exponential at small F in Eq.(21), accounting (22)
e
7
104 Eq.(21) τ
Eq.(23) ) 103 <mτp> 103
c
e τ, ε"(ω)
>103 es (s 100 <τ−1>−1 100
−1 m
><τ102 on ti10-3 10-3
<τ ati
ax10-6 10-6
101 el
r
10-9 10-9
0
10 0.05 0.1 0.15 0.2 0.25 0 10 20 30 0.7 0.8 0.9 1
δ F~ Fmp/kBT Tg/T
FIG. 4: The decoupling between the viscosity and the in- FIG. 5: Different relaxation times derived from the bar-
trinsicionicconductivity,asafunction oftherelativebarrier rier distribution in Eq.(23), as functions of the most prob-
e
width δF from Eq.(20), at Tg. The dashed and solid line ablebarrier(left)andthecorrespondingtemperature(right).
pertain to thespecific barrier distributions from Eq.(21) and δFe=0.25, corresponding to fragility D=4.
(23) respectively.
Note that the value of fragility used in Fig.5, D = 2,
We thus arrive at the following form:
is probably smaller than in CKN. In addition, we have
ignored here, for clarity, the effects of barrier softening
c1e−(1/Fe−1)2/2δFe2, F F
p(F)= Fe2 ≤ e (23) [27],thatwouldrequireintroducingasystem-specificad-
e e
( Fce22eF/(δF/4), Fee <eF ≤1, jsulosptaebsleofcotnhsetacnutrvTeAs.sTohmeelwahttaetr,ewffeitchtosuwtouaffldeccthiannggethtehier
e
where Fe and the normalization conestantes c1 and c2 are vertical separations.
chosensothatthedistributionisnormalized,continuous, To test the predictions from Figs.4 and 5, one needs
anditsfirstderivativeiscontinuoustoo. Thedistribution to know the width of the barrier distribution for α-
fromEq.(23)isplottedinFig.3. Thedecouplingstrength relaxation. As already mentioned, the gross features
τ τ−1 , computed for the composite distribution from of this distribution have been predicted by the RFOT
h i
Eq.(23), is shown in Fig.4, as a function of the relative theory, and have lead to quantitative predictions of the
(cid:10) (cid:11)
distribution width δF, at T . We therefore observe that correlation between the stretching exponent β and the
g
infragileliquids,theapparenttime-scaleseparationmay fragility D, and the deviations from the Stokes-Einstein
reachasmuchasfouerordersofmagnitude nearthe glass relation. The corresponding trends are as follows: more
transition - even though only one process is present! - fragileliquidsarepredictedtohavebroaderbarrierdistri-
because the inrinsic ionic conductivity is dominated by bution leading to a smaller value of β, and vice versa for
the fastest relaxing regions. stronger substances [3]. A correlation with the fragility
Conversely,whenthe apparentdecouplingexceeds the comes about by virtue of Eq.(20). Several a` priori ways
intrinsicvalueprescribedbyFig.4,wemayconcludethat to determine β and D have been employed, by experi-
ionicconductiondoesnotinfactrequirestructuralrelax- menters,that sometimes produce conflicting results. For
ation. This notion is of significance for the mechanisms example, the fragility extracted from τσ will be consis-
of electrical conductance in glasses and will be discussed tently lower than that extracted from the mechanical
in detail in the Conclusions. relaxation time τs, because τσ < τs. The exponent β
One may also illustrate the effects of apparent decou- from the stretched exponential is extracted from fits of
pling for a specific value of fragility, by plotting several various relaxation processes to a stretched exponential
varieties of relaxation times, as functions of the most profile e−(t/τ)β. Alternatively, one may choose to fit the
probable barrier, or the corresponding temperature, see Fouriertransformofthestretchexponential,ortheCole-
Fig.5. (Given the time scale at the glass transition, say Davidson form, to the imaginary part of ǫ(ω) in insula-
DT /(T T ) = ln(1016) 37 (see Eq.(3)), there is tors [39]. These usually produce comparable results for
K g K
− ≃
aone-to-onecorrespondencebetweenthe fragilityD and the corresponding exponent β, with a notable exception
the T /T ratio.) We observe that the average relax- ofionicconductors,whichhappentobethemainfocusof
g K
ation time and the one derived from the inverse of the this paper. In ionically conducting systems, the dc com-
maximum position of ǫ′′(ω) are close, and are near the ponentofthefulldielectricresponsefromEq.(18)largely
most probable value of the relaxation time. (The ǫ′′(ω) “swamps” the ac part so that reliable determinations of
was computed using Eqs.(18) and (23), see below.) The the latter are complicated. The reader is reminded that
apparent conductivity relaxation time is strongly decou- dielectricmeasurementsonionmeltsaredifficultbecause
pled, consistent with data of Howell at el. [11] for CKN. electrodes generally block ionic current. The effects of
8
8
F /k T=37 2222277777 1111177777 77777 3.0 ε"(ω)
6 mp B ω M"(ω)
ω) g
("4 lo32.5
ε 2 ∆1/
h,
0 widt 2.0
0.03 ak
ω) pe 1.5
( 0.02
"
M
1 1.1 1.2 1.3 1.4
0.01 T/T
g
0
-6 -3 0 3 6 9 12
10 10 10 10 10 10 10 FIG. 7: The widths of the peaks in the imaginary parts of
ω, sec-1 the dielectric constant ǫ(ω) and modulus M(ω), as functions
e
of temperature (c.f. Fig.6). δF =0.25.
FIG. 6: The top panel shows the imaginary part of the
non-conductivepart of the dielectric response, ǫ(ω),Eq.(17),
for four values of the most probable α-relaxation barrier: distribution from Eq.(23). We have used CKN’s values
e
Fmp/kBT = ln(τ/τmicro), as indicated on the graph. (δF = for ǫ0 and ǫ∞, as before. For the sake of argument, we
0.25, τ = 1 ps.) These barrier values were chosen be-
micro use δF =0.25, corresponding to β 0.4 at T . In Fig.6,
ctiatuasteivfeolyr τa/cτcmuricartoe,>weh7il≃e 1τ0/3τ, the R≃FOe3T7a≃pp1ro0a16chisisaqtuathne- bottom, we show the imaginary pa≃rt M′′(ω),gof the full
micro moduelus. Clearly the two functions exhibit qualitatively
upperlimitofthedynamicalrangeroutinelyaccessibleinthe
different behaviors. Note that the effect of the dc com-
lab. Thebottompanelshowthecorrespondingmodulus,from
Eq.(24), that includs the dc-part, due to the intrinsic ionic ponent on the apparent relaxation profile has been dis-
current. The same four values of the barrier are used. We cussed previously [17, 24], including the possibility of a
haveusedtheCKN’svaluesfor ǫ0 andǫ∞,andassumed that doublepeak[24]. ThelatterhasbeenobservedbyFunke
∆a3µk2mBoTl ≃2(ǫ′0−ǫ∞), see thediscussion preceeding Eq.(19). aelt.el[.37[]4.1]N, bevuetrthhaeslensost, tbheeendireelpercotrdiuccmedodbuyluPsimobentaoivneadt
here is qualitatively consistent with CKN’s data from
Ref.[37]. Finally note that for smaller dc-conductivities,
build-upchargeareoftentreatedphenomenologically,by
the modulus data would become more similar to ǫ′′(ω).
meansofequivalentcircuits[37,40]. Giventhesecompli-
We conclude from the above analysis that if one were
cations, many have chosen to plot the reciprocalof ǫ(ω),
to use the modulus data to extract the characteristics
i.e. the dielectric modulus [11, 40]:
of the barrier distribution, one must measure first the
M(ω) 1/ǫ(ω). (24) dc-current, add it to the ǫins(ω) from a microscopic the-
≡ ory, and then compare the result to the measured M(ω)
M(ω) is well behaved and even shows a peak in the data. But again, because of the large contribution of
imaginary component, similarly to ǫ(ω) of a near insu- the dc component, the corresponding fits would not dis-
lator. In the absence of an a` priori microscopic picture criminatewellbetweendifferentformsofǫ (ω). Onthe
ins
and by analogy with ǫ(ω), one might interpret this peak otherhand,treatingtheelectricfieldasaresponsetothe
as as the response of the electric field E to the dielec- displacement may lead to erroneous conclusions on the
tric displacement D. This in fact would be appropriate temperature dependence of the barrier width. In fact,
in a layered dielectric [11]. See also the discussions in the barrier widths derived from ǫ′′ (ω) or M′′(ω) show
ins
Refs.[16,17,18,19,20,21,22]. Yettheresultingvaluesof the opposite trends, as we have seen already in Fig.6.
the mostprobablerelaxationtime andthe stretchingex- One may further quantify this observation: In the ab-
ponent deviate from those obtained with other methods senceofamicroscopictheory,oneoftencharacterizesthe
[13, 19, 37]. In fact, the modulus-derived β increases, width of the ǫ′′ (ω) peak by a stretching exponent β,
M ins
whilethewidthoftheM′′(ω)peakdecreaseswithlower- as derived e.g. from Davidson-Cole fits. The distribu-
ing the temperature, in conflict with the general trends tion from Eq.(23) indeed gives rise a power law behav-
for poor conductors, and the conclusions of the RFOT ior, consistent with Eq.(22), see Appendix. In contrast,
theory. the correspondingM′′(ω)curvesdonotexhibitasimilar
The RFOT theory and the present results allow one power-lawbehavior. Ihavechosentoillustratetheoppo-
to address these difficulties, to which we devote the rest site temperature trends in the widths of ǫ′′ [ln(ω)] and
ins
of this Section. One first notes that structural reconfig- M′′[ln(ω)] peaks, by measuring the latter widths at one-
urations are compact, and so the layered-dielectric view third-height and plotting them as functions of temper-
of supercoold melts is not microscopically justified. We ature, see Fig.7. Similar opposite trends, too, would be
next plot, in the top panel of Fig.6, the non-conductive observedforthecorrespondingapparentbarrierwidthsor
ǫ′′ (ω),fromEq.(17)averagedwithrespecttothebarrier effective β’s. Clearly,interpreting the dielectric modulus
ins
9
ofanionicconductorasaresponsefunctionmayleadtoa
F /k T=37 27 17 7
significantunderestimationoftheactualbarrierwidthat 101 mp B
lowtemperatures,andqualitativelyincorrectconclusions
on the temperature dependence of the width.
)
ω
( 100
"
ε
V. CONCLUSIONS
-1
10
We have computed, from the first principles, the vis-
-6 -3 0 3 6 9 12
cosity andthe intrinsic ionic conductivity ofsupercooled 10 10 10 10 10 10 10
liquids. The viscosity is determined by four microscop- ω, sec-1
ically defined quantities: the length scale of the local
chemicalorderthatsetsinattemperatureT ,whereliq-
cr
uid dynamics become activated; the Lindemann length, FIG. 8: The four thick lines are the same as in Fig.6, but
characterizingmoleculardisplacementsatthemechanical in thedouble-log scale. Thethin linesare theDavidson-Cole
stability edge; the temperature; and the average relax- forms, following from a simpleapproximation, seeAppendix.
The dash-dotted line illustrates how the stretching exponent
ation time τ of the activated reconfigurations that dom-
β was extracted from thecurves.
inate the liquid dynamics below T . The extraordinar-
cr
ily long τ range is what gives rise to the high viscos-
ity of the liquid when it approaches the glass transition.
ceedingtheLindemannlengthinatimeittakesthelocal
When the local chemically stable units (or “beads”) are
environmenttorelax. Therefore,localrelaxationisnota
charged, the fluid will also exhibit an ionic conductiv-
necessary condition for a non-zero current of these ions.
ity, which we have called the “intrinsic” conductivity, to
Some interaction with relaxation may still be present,
constrastit with electricconductionvia delocalizedelec-
however at large enough decouplings, we may say that
tronic carriers or via mobile ions that are not bonded
the ion (or any other carrier) interacts with the liquid
to the metastable aperiodic lattice forming the super-
as if the latter were a perfectly stable, albeit disordered
cooled liquid. Computing the conductivity requires an
lattice. In such cases,one may think of the ionic current
additional microscopic characteristic, the electric charge
in superionic conductors in terms of regular, not slaved
ona“bead”. Fortunately,thisadditionalparametermay
diffusion. In regular diffusion, the total travel time is
bededucedfromtheacdielectricresponse,whichwehave
dominated by the slowest step, in contrast to Eq.(15).
also estimated. Perhaps the main finding of this work is
The intrinsic difficulty in experimental assessment of
thatincontrastwiththeviscosity,theionicconductivity
the barrier distribution in moderately conductive melts
isdominatedbythefastestrelaxingregionsintheliquid,
is thatthe dc currentdominatesthe overalldielectric re-
as reflected in Eq.(15).
sponse. This gives rise to ambiguities as to what the
We have discussed ways to test the above predictions,
actualwidth ofthe barrierdistribution is, since mechan-
the most important aspect of which is the large separa-
ical relaxation and dielectric modulus data disagree. We
tion, or “decoupling”, between the apparent time scales,
have shown that this is expected, and argued that the
suggested previously by viscosity and ionic conductiv-
mechanical relaxation offers the preferred method of es-
ity data on purely phenomenological grounds. We have
timating the actual barrier distribution.
shownthatsuchapparenttime-scaleseparationisindeed
Acknowledgments: The author thanks Peter G.
expected because of the very broad barrier distribution
Wolynes for critical comments and useful suggestions.
forα-relaxation,derivedearlierinthe RandomFirstOr-
He gratefully acknowledgesthe GEAR, the New Faculty
der Transition (RFOT) methodology. The decoupling
Grant, and the Small Grant Programs at the University
thus stems essentially from the same cause as the vio-
of Houston.
lation of Stokes-Einstein relation in supercooled liquids
[4]. Now, we have seen that the value of the decoupling
isnotverysensitivetothepreciseformofthebarrierdis-
tribution so long as one acounts for the RFOT-derived Appendix
grosscharacteristicsofthis. We havethus quantifiedthe
degree of “decoupling”: The intrinsic ionic conductivity Let us see that the distribution in Eq.(23) is qualita-
was arguedto decouple at mostby four ordersofmagni- tively consistent with experimental ǫ (ω) and the em-
ins
tudefromthelow-frequencymomentumtransport. Con- pirical correlation of β and D. For this, we replot the
versely, any conductivity exceeding this limit must be top panel of Fig.6 in the double-log format, in Fig.8.
duetootherchargecarriersthatdonotdisturbtheliquid We note the generaladequacy of the barrierdistribution
structure beyond typical vibrational displacements. In- fromEq.(23): Similarlytothe experimentalǫ(ω)inpoor
deed,supposethe apparentdecouplingexceedsthe value conductors, the resulting high-frequency wing is signif-
prescribed by the width of the barrier distribution. This icantly broader than the low-frequency one. Note that
means that there will be ions that travel a distance ex- the actual data would also often display an additional
10
1
fied by the dash-dotted line in Fig.8. The dependence
of the thus obtained exponent β on the fragility D, at a
0.8 fixedFmp/kBT =37,is shownby the dashed-dottedline
in Fig.9. This β is, again, qualitatively consistent with
experiment. Greater accuracy should not be expected
0.6
here, as we have not treated the higher-frequency range
β
associated with β-relaxation, which would affect the ex-
0.4
perimentally determined stretching exponents.
In addition, we verify that the informal argument in
0.2 Eq.(25) the main text that the width of the barrier distribution
ε"(ω) should be about δF/4, at the half-height or so. Indeed,
0 for a gaussian barrier distribution with width δF/4 =
0 2 4 6 8 10 12
1/2 F/8√D impliesthefollowingapproximateexpressionfor
D
the stretching exponent β at T (c.f. Eq.(9) of Ref.[3]):
g
FIG.9: Twoapproximaterelationsofthestretchingexponent
βtothefragilityDfromtheVogel-Fulcherform,fromEq.(25) F/k T 2 −1/2
B
and as derived from the slopes of the high-frequency wing of β = 1+ , (25)
theǫ′′(ω) peaks, such as in Fig.8. " (cid:18) 8√D (cid:19) #
shown as the solid line in Fig.9. This expression is in
high-frequency wing, which is ascribed to the secondary, very good agreement with experiment, see Fig.2 from
β-processes, also called Johari-Goldstein relaxation [42]. Ref.[3]. (At T on scale τ/τ = 1016, F/k T 37.)
g micro B
≃
(See [43] for a review). The present results suggest that Note also Eq.(25) is consistent with Eq.(20), assuming
β-relaxation does not contribute to the intrinsic ionic the Davidson-Cole [44] and William-Watts [45] stretch-
conductivity. At any rate, the derived ǫ(ω) show sev- ing exponents β are close [39]. That the latter is the
eral decades of nearly power-law decay, allowing one to caseindeedIdemonstratebygraphingtheDavidson-Cole
extract the corresponding exponent: ǫ(ω) ω−β. The (DC) form (ǫ (ω) ǫ ) = (ǫ ǫ )(1 iωτ)−β [44]
DC ∞ 0 ∞
effective β’s were deduced from the slopes ∝of the curves with τ =τmicroeFmp−/kBT and β fr−om Eq.(2−5). These are
at the points ofmaximum secondderivative,as exempli- shown in Fig.8 as thin dashed lines.
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